To determine which of the given expressions represents the volume of a cylinder, we must recall the formula for the volume of a cylinder. The volume [tex]\( V \)[/tex] of a cylinder with radius [tex]\( r \)[/tex] and height [tex]\( h \)[/tex] is given by:
[tex]\[ V = \pi r^2 h \][/tex]
In this context, suppose the radius [tex]\( r \)[/tex] is denoted by [tex]\( x \)[/tex] and the height [tex]\( h \)[/tex] is also denoted by [tex]\( x \)[/tex]. Then, the volume formula becomes:
[tex]\[ V = \pi x^2 \cdot x \][/tex]
[tex]\[ V = \pi x^3 \][/tex]
We need to identify which of the given expressions matches the form [tex]\( \pi x^3 \)[/tex].
1. [tex]\( 3 \pi x^2 + 4 \pi x + 16 \pi \)[/tex]
- This expression does not match [tex]\( \pi x^3 \)[/tex]; it contains terms with [tex]\( x^2 \)[/tex], [tex]\( x \)[/tex], and a constant term.
2. [tex]\( 3 \pi x^2 + 16 \pi \)[/tex]
- This expression also does not match [tex]\( \pi x^3 \)[/tex]; it consists of terms with [tex]\( x^2 \)[/tex] and a constant.
3. [tex]\( 3 \pi x^3 + 32 \pi \)[/tex]
- This expression includes a term [tex]\( 3 \pi x^3 \)[/tex] plus a constant term. Notice that the coefficient of [tex]\( x^3 \)[/tex] is 3, not 1. While it contains [tex]\( \pi x^3 \)[/tex], it is actually multiplied by 3.
4. [tex]\( 3 \pi x^3 + 20 \pi x^2 + 44 \pi x + 32 \pi \)[/tex]
- This expression contains multiple terms with different powers of [tex]\( x \)[/tex] and does not match the simple form [tex]\( \pi x^3 \)[/tex].
Among the given expressions, option 3, [tex]\( 3 \pi x^3 + 32 \pi \)[/tex], includes the term [tex]\( 3 \pi x^3 \)[/tex]. It suggests a connection to the volume formula [tex]\( \pi x^3 \)[/tex], albeit scaled by a constant.
Thus, the expression that includes the term matching the volume formula under given assumptions is:
[tex]\[ 3 \pi x^3 + 32 \pi \][/tex]
Therefore, the correct answer is:
[tex]\[ 3 \pi x^3 + 32 \pi \][/tex]
Which corresponds to the third option.
